Domain patterns in single crystal ferroelectrics
commonly take the form of nearly periodic laminations [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}.
We can define a rank-1 laminate, as a pattern comprising alternating
layers of two domain types, each of which is a single crystal variant.
Examples include alternating bands of \(180^{\circ }\)  or \(90^{\circ }\) 
domains. Similarly, simple laminations can themselves be layered together
to produce a higher rank laminate. Based on the linear constrained
theory of laminates, Tsou et al.
[8]} identified several rank-2 periodic domain patterns
in tetragonal ferroelectrics. Of these, a layered stripe domain pattern
and herringbone domain pattern, are both well-known and commonly observed
[9]}, [10]}, [6]}, [12]}. Meanwhile, certain other patterns
are rarely seen in experiments [13]}, [14]}, though elements
of these patterns sometimes appear [2]}. In the present
work, we consider periodic polarization patterns formed in macroscopic
single crystals, but with nanoscale domains. These polarization patterns
are referred to as nanoscale periodic patterns. The theory employed
by Tsou et al.
[8]} neglected gradient energy due to domain walls and the
elastic/dielectric energy due to disclinations generated at the junctions
of domains. However, these factors could be expected to be of importance,
especially in nanoscale periodic patterns [17]}, [18]}. This
leads us to consider the following questions: Among the periodic laminates
identified by the constrained theory, which are stable when gradient
and elastic energy are considered? Which patterns constitute global
energy minima, and does this depend on the scale? Can imposed states
of average strain or polarization stabilize specific patterns, allowing
the nano-engineering of domain patterns? 
